The most accepted theory of gravitation is general relativity (GR) of Einstein. The reason for this is the good agreement of the results of the theory with experiments for weak fields where only low order approximations of GR are needed. GR gives for homogeneous, isotropic, cosmological models a singularity of the solution in the beginning with infinite density of matter, the so-called big bang. This demands the expansion of space with cosmic inflation after the singularity to get agreement with the experimentally verified flat space.
In this article, the theory of gravitation in flat space-time (GFST) is applied to cosmological models. It gives for weak gravitational fields the same results as GR to measurable accuracy (see  ). Let us study by the use of GFST homogeneous, isotropic, cosmological models in the pseudo-Euclidean metric, i.e. space is flat. In the beginning of the universe all the energy is gravitational energy, i.e. no matter. In the course of time gravitational energy is converted to matter without contraction or expansion of space. There are non-singular, cosmological models, i.e. no big bang. The result is a non-stationary universe with conservation of the total energy. The redshift is a gravitational effect and not a Doppler effect. The solution is non-singular, i.e. matter is always finite and no big bang exists. The interpretation of the result in an expanding universe is possible because of the non-singular solution and the general covariance of GFST.
2. Homogeneous, Isotropic, Cosmological Model
Let us summarize the results of homogeneous, isotropic, cosmological models of GFST. GFST is studied in the book  and in many articles, e.g.    and  where homogeneous, isotropic, cosmological models are given, too. GFST was first considered in article  where the metric is the pseudo-Euclidean geometry. The theory was applied to cosmology yielding non-singular solutions (see article  ). A non-expanding universe was already considered in article  .
We follow along the lines of article  by the use of the pseudo-Euclidean metric, i.e.
with and the Cartesian coordinates . The pseudo-Euclidean metric is the standard geometry for physical problems and implied by astrophysical observation of flat space. The universe is described by the use of the gravitational field
with the proper-time τ:
Let ( ) be the four-velocity with
which implies by (2.1) and (2.2)
The matter tensor is a perfect fluid with pressure p and density ρ and
where m and r denote matter and radiation. The equations of state are
Let be the present time and assume the initial conditions at present:
where is the Hubble constant, is a constant which doesn’t appear by GR and and denote the present densities of matter and radiation of the universe. It follows under the assumption that matter and radiation do not interact
GFST gives two field equations for a and h and the conservation of the total energy. It follows by suitable combinations of these equations and integration by the use of the initial conditions (2.5) (see e.g.   )
and is the total conserved energy. Let , and be the standard definitions of the density parameters of matter, radiation and vacuum energy and put
The field equations of gravitation give two differential equations of order two and one equation by the use of the conservation of the total energy. Suitable combinations of these three differential equations imply by longer calculations (see article  ) the differential equation
with the initial condition
The solution of (2.10) together with (2.7) describes a homogeneous, isotropic, cosmological model by the use of GFST. Furthermore, it holds
A necessary and sufficient condition to avoid singularities of the solution of (2.10) is (see e.g.   )
for all . Hence, condition (2.12) implies a non-singular solution for all , i.e. we get a non-singular, cosmological model.
It exists such that
Put then it follows from (2.10a) with
It holds for all
Let us assume
then we get by the use of (2.14)
It follows from (2.9) that the sum of the density parameters is a little bit greater than one
The presently assumed density parameter of matter is
It follows from (2.9) and (2.17)
with the assumption that the sum of these density parameters is a little bit greater than one, i.e. we get the same approximation as by GR but with different assumptions, namely flat space by GR and no singularity by GFST.
We get from (2.10) and (2.7) that starts for from a small positive value, decreases to the small positive value and then increases for all whereas starts from infinity, decreases to a positive value and then increases (see e.g.   ).
The differential equation (2.10a) is rewritten by the use of (2.7) in the form
The proper-time from the beginning of the universe till time is
The time t starts from whereas the proper-time starts from zero and then increases till , i.e. the age of the universe. Hence, the age of the universe is finite measured with proper-time and the age is infinite measured system time given by the pseudo-Euclidean metric. Therefore, a time before the proper-time of the age of the universe makes no sense.
The differential Equation (2.18) is rewritten by the introduction of the proper-time :
This differential equation is identical with that of GR with zero curvature by virtue of (2.17) for and a sufficiently large. Therefore, the result for the universe of a flat space of GR for sufficiently large a, i.e. sufficiently large away from the singularity is nearly identical with that of GFST enough time after time . It worth to mention that the beginning of the universe by GFST and by GR are quite different. The cosmological model of GFST contains in the beginning of the universe only gravitational energy and no matter. In the course of time matter, radiation and vacuum energy is generated where the total energy is conserved. Gravitational energy is partly converted to matter and other energies. There is no contraction and no expansion of space by virtue of the pseudo-Euclidean metric (2.1) implying a flat space. A flat space is also received if we use (2.2) as metric as by GR.
It is worth to mention that the condition (2.12) is important to avoid a singularity. GR gives (2.20) with which implies the singularity, i.e. the big bang. The detailed calculations of these results can be found in the book  .
is the time measured with atomic clocks but for mathematical calculations it is sometimes useful to introduce the absolute time by
where denotes the Euclidean norm. Light velocity is equal to vacuum light velocity.
The differential Equation (2.18) is rewritten by the use of the absolute time of (2.21) in the form
The energy of a photon emitted at time from an object at distance r from the observer is by virtue of (2.21),
This gives for the energy of the photon
where is the emitted energy of the same atom at time . The energy of the photon which moves to the observer in the universe is by virtue of (2.22) constant (see e.g.  ). Then, the corresponding frequency is
where is the received frequency by the observer and the frequency emitted from the same atom at the observer. The redshift is by definition the difference of the observed wavelength and the standard one divided through the standard one. Hence, the redshift satisfies the relation (expressed by the corresponding frequencies)
Light emitted at distance r at time and received at at time satisfies by the constant velocity of light the relation
This implies by Taylor expansion of of (2.25)
Differentiation of equation (2.23) gives by neglecting small expressions
Hence, the redshift has the form
The detailed calculation of this result can be found in the book  . The redshift is already derived in article  without use of the Doppler effect. This cosmological model was already studied in article  where as well as the interpretation of a bounce is used and the interpretation of a non-expanding universe is mentioned.
GFST is a covariant theory of gravitation in flat space-time. It gives for weak gravitational fields to measurable accuracy the same results as GR. But there are also differences to the results of GR for strong gravitational fields, e.g. in the beginning of the universe The gravitational field is described by non-linear differential equations of order two in divergence form where the source is the total energy-momentum tensor including that of gravitational field. The conservation of the total energy-momentum tensor gives the equations of motion of matter. The theory is generally covariant. The space of the universe is flat. The universe starts with uniformly distributed gravitational energy in the whole space, i.e. no matter and no other forms of energy. In the course of time matter, radiation and vacuum energy are created where the total energy of the universe is conserved. Neither expansion nor contraction of space takes place. Singularities don’t arise, i.e. all physical quantities are finite and big bang doesn’t exist. The redshift of spectral lines of distant objects is a gravitational effect by virtue of the time dependent gravitational field. The age of the universe is finite measured with proper-time but infinite measured with system time which is given by the pseudo-Euclidean metric.